CSDC Seminar: Harmonic analysis and boundary value problems in geometry

Historically, boundary value problems have appeared in engineering problems as partial differential equations on structures with boundary. However, in the past half century, differential operators have played a crucial role to encode, understand and resolve geometric and topological questions. This often requires the deformation of boundary conditions and harnesses index theory as a control mechanism through utilising the invariant nature of the index. However, such deformations arguments typically requires an understanding of all possible boundary conditions. For Dirac-type operators, which are often seen in geometry, this is now well understood. However, not all geometric operators are of Dirac-type. The quintessential example is the Rarita-Schwinger operator, which incidentally captures the physics of the Delta baryon, a type of subatomic particle.

In this talk, I will present some motivating examples from geometry to illustrate the significance of  boundary value problems in geometry. I will showcase recent contributions made by myself and collaborators to the study of boundary value problems for first-order elliptic operators. In particular, this includes the Rarita-Schwinger operator as a special case, moving far beyond the Dirac-type regime that was historically studied. A key novelty in our analysis is the use of modern harmonic analytic methods, which I will briefly outline. I will also illustrate some wider consequences of these methods in geometric applications. The talk will conclude with a select description of ongoing and future work.